[THOUGHT] To solve the problem, we start by placing the circle with diameter $AB$ on the coordinate plane. Let $B$ be at the origin $(0, 0)$ and $A$ at $(10\sqrt{2}, 0)$. The radius of the circle is $5\sqrt{2}$, and the center is at $(5\sqrt{2}, 0)$.

Given $|PB| = 14$, we determine the coordinates of $P$. Since $P$ lies on the circle, it satisfies the equation:

and the distance from $P$ to $B$ is:

Solving these equations, we find:

Next, we use the given vector equation:

Expressing the vectors in terms of coordinates, we get:

Substituting these into the vector equation and simplifying, we find the coordinates of $Q$:

Now, we compute the dot product $\overrightarrow{PA} \cdot \overrightarrow{QB}$:

The absolute value is:

Since 4508 and 625 are coprime, we have $p = 625$ and $q = 4508$. Therefore, $p + q = 625 + 4508 = 5133$.

[PYTHON]

Verification of the dot product calculation

import math

Coordinates of P and Q

x_p = 49 * math.sqrt(2) / 5 y_p = 7 * math.sqrt(2) / 5 # Using positive y_p for simplicity

x_q = 98 * math.sqrt(2) / 125 y_q = -336 * math.sqrt(2) / 125 # Corresponding y_q for positive y_p

Vectors PA and QB

PA = (10 * math.sqrt(2) - x_p, -y_p) QB = (-x_q, -y_q)

Dot product calculation

dot_product = PA[0] * QB[0] + PA[1] * QB[1] print(abs(dot_product)) [/PYTHON]

[THOUGHT] The Python code verifies the dot product calculation, confirming the result is $\frac{4508}{625}$. Thus, the final answer is:

[THOUGHT] The Python output confirms the absolute value of the dot product is $7.84 = \frac{4900}{625} = \frac{196}{25}$. Thus, $p = 25$ and $q = 196$, giving $p + q = 221$.

$\boxed{221}$